Question 6.8.1: Find dy/dx when: (a) y = u^5 and u = 1 − x³ (b) y = 10/(x² +......

Essential Mathematics for Economic Analysis [1185377]

Find dy/dx when:

(\mathbf{a})~~y=u^{5}{~\mathrm{and}~u}=1-x^{3}~~~~~~~~~~~~~~~~(\mathbf{b})~~y=\frac{10}{(x^{2}+4x+5)^{7}}

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(a) Here we can use (6.8.1) directly. Since $dy/du = 5u^{4}$ and $du/dx = −3x^{2},$ we have

{\frac{\mathrm{d}y}{\mathrm{d}x}}={\frac{\mathrm{d}y}{\mathrm{d}u}}\cdot{\frac{\mathrm{d}u}{\mathrm{d}x}}=5u^{4}(-3x^{2})=-15x^{2}u^{4}=-15x^{2}(1-x^{3})^{4}

(b) If we write $u = x^{2} + 4x + 5,$ then $y = 10u^{−7}.$ By the generalized power rule (6.8.2), one has

$y^{\prime}=a u^{a-1}u^{\prime}$ (6.8.2)

{\frac{\mathrm{d}y}{\mathrm{d}x}}=10(-7)u^{-8}u^{\prime}=-70u^{-8}(2x+4)={\frac{-140(x+2)}{(x^{2}+4x+5)^{8}}}

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